Relations

More info about relations:

• The sets $S$ and $T$ that create a relation might differ from one another regarding the kinds of elements they have. Example: $\text{CUNYfirst-Grades}$ from before is a relation from numerical grades (= integers from $0$ to $100$) to letter grades. You may also create relations from some familiar sets, e.g., from $\mathbb{Z}$ to $\mathbb{N}$, such as a relation of an integer to its absolute value: $\text{Abs} = \;$$\{(-5, 5), \,$$(-3, 3), \,$$(0, 0), \,$$(2, 2), \,$$(3, 3), \dots\}$. We can, therefore, write that $-5\ \text{Abs}\ 5$ and $3\ \text{Abs}\ 3$, for instance.
• You may even relate a set $S$ to itself. Example: The relation $\text{Equals}$, or '$=$', from $\mathbb{P}$ to $\mathbb{P}$ is one that relates a positive integer to itself: $\text{Equals} = \;$$\{(1, 1), \,$$(2, 2), \,$$(3, 3), \,$$(4, 4), \dots\}$. We can, therefore, write that $2\ \text{Equals}\ 2$, or $2 = 2$, for instance. A relation of a set $S$ to itself is shorthandedly called a relation on $\boldsymbol S$.
• An example of a relation on $\boldsymbol{\mathbb{N}}$: The $\text{LessThanOrEqualTo}$ relation, or '$\le$', is as follows:
  $\text{LessThanOrEqualTo} = \;$$\{(0, 0), \,$$(0, 1), \,$$(0, 2), \dots, \,$$(1, 1), \,$$(1, 2), \,$$(1, 3), \dots, \,$$(2, 2), \,$$(2, 3), \dots\}$. For example, we see that $1 \le 1$ and that $2 \le 3$.
  Bonus question: What's the idea of this relation? E.g., given the pattern above, to which integers will $\boldsymbol 5$ relate?